Groebner Fans in Combinatorics, Representation Theory and Commutative Algebra: Theory and Computation

University Of Washington, Seattle WA

Investigators

Abstract

Abstract for award DMS-0401047 of Thomas: Groebner basis theory is today a central theoretical and computational tool in algebraic geometry and commutative algebra with applications in myriad fields such as computer science, optimization, robotics, statistics, and computer modeling. The Groebner fan of an ideal or module is the decomposition of the space of weight vectors that induce Groebner base, such that each cone in the fan indexes a distinct Groebner basis. This project describes four problems from combinatorics, optimization, representation theory and algebraic geometry that revolve around Groebner fans. The first project investigates the complexity, geometry and homological properties of initial ideals of toric ideals. Answers to these questions have implications in integer programming and on the computation of toric Groebner bases which themselves have several applications. The next examines the variation of weight vectors in Kalai's theory of algebraic shifting for simplicial complexes. The third project studies the polyhedral geometry of moduli of resentations of the McKay quiver which (partially) resolve singularities that arise in generalized McKay correspondence. The last goal is to develop a software package for computing Groebner fans of arbitrary polynomial ideals. Applications include the computation of tropical varieties and the study of orbit closures and degenerations of representations of finite dimensional algebras. The proposed work will employ tools from combinatorics, discrete geometry and optimization to solve problems from commutative algebra, representation theory and algebraic geometry. All projects have a strong computational component that aims at effective algorithms and serves as a laboratory for uncovering new structure theorems. The collaborations include three graduate students and three current or recent postdocs.